Geodesic
On the equation $\mathbf{h}x - x \mathbf{k}=c$ in Banach algebra $A$ with Hermitian coefficients $\mathbf{h},\mathbf{k}$
Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2002), pp. 3-8

Voir la notice de l'article provenant de la source Math-Net.Ru

This work investigates the problem of solvability on the equation $\mathbf{h}x - x\mathbf{k}=c$ with Hermitian coefficients $\mathbf{h},\mathbf{k}$ of weak complete Banach algebra $A$. In the article (theorem 1) the criterion of solvability of equation $\mathbf{h}x-x\mathbf{k}=c$ is proved, which implies (theorem 2) the following algebraic criterion of solvability. For solvability in $A$ the equation $\mathbf{h}x-x\mathbf{k}=c$ the similarity of matrixes $\left(\begin {array}{cc}\mathbf{h}, 0 \\ 0, \mathbf{k} \end {array} \right)$ and $\left(\begin {array}{cc}\mathbf{h}, 0 \\ c, \mathbf{k} \end {array} \right)$ is necessary and sufficient.
Keywords: solvability on the equation, Banach algebra.
Mots-clés : Hermitian coefficients 
I. M. Karakhanyan. On the equation $\mathbf{h}x - x \mathbf{k}=c$ in Banach algebra $A$ with Hermitian coefficients $\mathbf{h},\mathbf{k}$. Proceedings of the Yerevan State University. Physical and mathematical sciences, no. 1 (2002), pp. 3-8. http://geodesic.mathdoc.fr/item/UZERU_2002_1_a0/
@article{UZERU_2002_1_a0,
     author = {I. M. Karakhanyan},
     title = {On the equation $\mathbf{h}x - x \mathbf{k}=c$ in {Banach} algebra $A$ with {Hermitian} coefficients $\mathbf{h},\mathbf{k}$},
     journal = {Proceedings of the Yerevan State University. Physical and mathematical sciences},
     pages = {3--8},
     year = {2002},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/UZERU_2002_1_a0/}
}
TY  - JOUR
AU  - I. M. Karakhanyan
TI  - On the equation $\mathbf{h}x - x \mathbf{k}=c$ in Banach algebra $A$ with Hermitian coefficients $\mathbf{h},\mathbf{k}$
JO  - Proceedings of the Yerevan State University. Physical and mathematical sciences
PY  - 2002
SP  - 3
EP  - 8
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/UZERU_2002_1_a0/
LA  - ru
ID  - UZERU_2002_1_a0
ER  - 
%0 Journal Article
%A I. M. Karakhanyan
%T On the equation $\mathbf{h}x - x \mathbf{k}=c$ in Banach algebra $A$ with Hermitian coefficients $\mathbf{h},\mathbf{k}$
%J Proceedings of the Yerevan State University. Physical and mathematical sciences
%D 2002
%P 3-8
%N 1
%U http://geodesic.mathdoc.fr/item/UZERU_2002_1_a0/
%G ru
%F UZERU_2002_1_a0

[1] M. A. Naimark, Normirovannye koltsa, Nauka, M., 1968 | MR | Zbl

[2] J. F. Bonssal, J. Duncan, Complete Normed Algebras, Springer-Verlag, 1973 | MR | Zbl

[3] P. Khalmosh, Gilbertovo prostranstvo v zadachakh, Mir, M., 1970 | MR | Zbl

[4] Yu. L. Daletskii, M. G. Krein, Ustoichivost reshenii differentsialnykh uravnenii v banakhovom prostranstve, Nauka, M., 1970, 534 pp. | MR | Zbl

[5] M. Rosenblum, Proc. Amer. Math. Soc., 20 (1969), 115–120 | MR | Zbl

[6] F. R. Gantmakher, Teoriya matrits, Nauka, M., 1988 | MR | Zbl

[7] E. A. Gorin, “Estimation of a partial involution in a Banach algebra”, Russian J. Mathematical Physics, 5:1 (1997), 117–118 | MR | Zbl